Thursday, December 29, 2005

A New Look at the Finite Element Method (FEM) by DLN Sastry UUTM 1969

A New Look at the Finite Element Method (FEM) By DLN Sastry29th October 2002=========================================

The classical FEM is (over)used in many continuous domain problems. Robust commercial software is currently available for solid stress analysis, fluid flow, heat transfer, electromagnetism, and so on. Significant advances have taken place in improving computational accuracy, speed and range of applications.

Without introducing much jargon and equations, additional possibilities of the method are shown here to exist. This is achieved by bringing in concepts of hierarchical and multilevel optimisation theory. Hierarchical and multilevel methods were originally developed for analysing complex systems and later applied to problems in specific fields like control, estimation and optimisation. Basically, the method involves first "decomposing" a large complex interconnected system into small manageable ones. These subsystems are then analysed independently of one another and their interconnections are restored by creating a "coordinator". Thus, several conveniently-sized subsystems are analysed at the lower level and their actions are coordinated by a higher -level coordinator. Such systems are numerically more robust and conceptually simpler.

Many problems analysed by the finite-element method are formulated as minimisation problems. The entity that is minimised is the energy, potential or something similar. The hierarchical formulation lends itself readily to such problems. With this introduction and without going into the detailed mechanics of the method itself, a few applications are pointed out below where the hierarchical multilevel finite-element method has been or can be applied very effectively.

1. Non-linear problems: These problems require an iterative solution and, hence, are generally computationally intensive and prone to numerical instability. While the effort is still more than a linear problem, a hierarchical solution is both considerably more stable and faster than a single-level solution.

2. Adaptive mesh: The standard method of increasing the solution accuracy while keeping a check on the effort is to distribute the mesh so that it is dense in the "critical" areas and sparse elsewhere. A two-step solution is commonly available in commercial software where, for example in stress analysis, the energy density is first minimised to obtain an "optimal" mesh. Theoretically, the true optimal mesh can only be achieved by iterating between the analysis and adaptive steps. In practice, a single iteration is used. Using the multilevel approach, a joint formulation can be used where the two steps are combined into one. While it would be horrendous in a single-level approach, the decomposition-coordination scheme can achieve significant savings in effort.

3. Design optimisation: The design optimisation problem includes a full analysis at each design modification step. For large design problems, the effort is too much to attempt complete optimisation . Often, a simpler (sub)optimal design is achieved by restricting the scope of design changes, quality of optimisation or some other simplification. Instead, the integrated problem may be cast into a bi-criterion joint optimisation problem and solved by a three-level decomposition-coordination scheme.

4. Discontinuities: This is an area where the single-level finite-element method gets into serious problems. The adaptive mesh facility tends to allocate a highly refined mesh in the vicinity of discontinuity but no matter how refined, the discontinuity tends to have a finite thickness. A true discontinuity is just a surface. Obviously, if the shape and location of the discontinuity are known a priori, the problem is very considerably simplified, but there are several problems where this is not the case. Examples are, transonic flow (sonic surface), supersonic flow (shock surface), fracture mechanics (fracture location?), etc where the unknown entities are inside parentheses. A two- or three-level formulation is very effective in determining the unknown location and analysing the resultant field at the same time.

5. Coupled problems: Interaction problems arise where two media are in contact. Examples are liquid slosh in bladder tanks, aerodynamics of sails and hang gliders, lateral jets from slender bodies (to serve as high lift devices) etc, These problems are characterised by unknown geometry and hence, the formulation involves coupled and non-linear equations; the solution is not always well-behaved if iterated between the geometric and physical variables. Again, a decomposition-coordination solution can come to the rescue.

These are meant to give a flavour for the method and demonstrate the power and versatility of the approach. Possibly there are many other areas where the paradigm would be handy.